#13155059

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

C₃³ ⋊ #350

	subgroup	G/H
center, Frattini, hypercenter	C₈	#157⋊D₃
commutator	C₃×C₃⋊Dic₃	C₂×C₈
Fitting	C₃³×C₈	D₄
socle	C₂×C₃³	C₂²⋊C₈
prime radical	#89042	C₄
cosocle	C₃×C₈×C₃⋊D₃	C₂²
hypocenter	C₃³	#350
LCS	G ▷ C₃×C₃⋊Dic₃ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₈

232 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

μ_t(G) = 96

Polycyclic presentation:

2^6,3^3|b^2=cdef,c^2=f,d^2=e,e^2=f,b^a=bc,c^a=cf,g^a=g^2,h^a=h^2,h^b=i,i^b=h^2,h^c=h^2,i^c=i^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 21122);